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# Introduction to Complex Numbers

when we square a real number it is impossible to get a negative real number.

Introduction to Complex Numbers

Before introducing complex numbers, let us try to answer the question ŌĆ£Whether there exists a real number whose square is negative?ŌĆØ LetŌĆÖs look at simple examples to get the answer for it. Consider the equations 1 and 2. This is because, when we square a real number it is impossible to get a negative real number.

If equation 2 has solutions, then we must create an imaginary number as a square root of ŌłÆ1. This imaginary unit ŌłÜŌłÆ1 is denoted by i .The imaginary number i tells us that i2 = ŌłÆ1. We can use this fact to find other powers of i

## Powers of imaginary unit i We note that, for any integer n , in  has only four possible values:  they correspond to values of n when divided by 4 leave the remainders 0, 1, 2, and 3.That is when the integer n Ōēż ŌłÆ4  or  n Ōēź 4 , using division algorithm, n can be written as n = 4q + k, 0 Ōēż k < 4, k and q are integers and we write

i)n = (i)4q+k = (i)4q(i)k = ((i)4)q(i)k =  (i) q(i)k =(i)k

Example 2.1

Simplify the following

(i) i7 (ii) i1729 (iii) i ŌłÆ1924 + i2018 (iv) (v) i i2 i3 ŌĆ”.. i40

Solution

(i) (i)7 = (i)4+3 = (i)3 = -i;

(ii) i1729 = i1729i1 = i

(iii) (i)-1924 + (i)2018 = (i)-1924+0 + (i)2016+2 = (i)0 + (i)2 = 1-1 = 0

(iv) = (i1 + i2 + i3 + i4) + (i5 + i6 + i7 + i8 ) + ŌĆ” + (i97 + i98 + i99 + i100 +) + i101 + i102

= (i1 + i2 + i3 + i4) + (i1 + i2 + i3 + i4) + ŌĆ” + (i1 + i2 + i3 + i4) + i1 + i2

= {i+(-1)+(-i)+1} + {i+(-1)+(-i)+1} + ŌĆ”. ŌĆ” + {i+(-1)+(-i)+1} + i + (-1)

= 0 + 0 + ŌĆ” 0 + i -1

= ŌłÆ1 + i (What is this number?)

(v) i i2 i3 ŌĆ” i40 = i1+2+3+ŌĆ”+40 = i [40x41]/2 = i820 = i0 = 1

Note

(i) ŌłÜ[ab] = ŌłÜaŌłÜb valid only if at least one of a, b is non-negative.

For example, 6 = ŌłÜ36 = ŌłÜ[(-4)(-9)] = ŌłÜ(-4) ŌłÜ (-9) = (2i) (3i) = 6i2 = -6, a contradiction.

Since we have taken ŌłÜ[(-4)(-9)] = ŌłÜ(-4) ŌłÜ (-9), we arrived at a contradiction.

Therefore ŌłÜ[ab] = ŌłÜa ŌłÜb valid only if at least one of a b, is non-negative.

(ii) For y Ōłł R , y2 Ōēź 0 iy = yi.

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12th Mathematics : UNIT 2 : Complex Numbers : Introduction to Complex Numbers |